Ace Maths Unit Two: Developing Understanding in Mathematics (word)

In this unit, the theoretical basis for teaching mathematics – constructivism – is explored. Varieties of teaching strategies based on constructivist understandings of how learning best takes place are described

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Unit Two: Developing Understanding in MathematicsContentsHow the unit fits into the module.............................................................................................................1 Overview of content of module.........................................................................................1 How this unit is structured.................................................................................................3Unit Two: Developing Understanding in Mathematics .........................................................................4 Welcome ...........................................................................................................................4 Unit outcomes....................................................................................................................6 Introduction.......................................................................................................................7 A constructivist view of learning.......................................................................................8 The construction of ideas..............................................................................10 Implications for teaching..............................................................................13 Examples of constructed learning.................................................................13 Construction in rote learning........................................................................15 Understanding...............................................................................................16 Examples of understanding...........................................................................18 Benefits of relational understanding.............................................................21 Types of mathematical knowledge..................................................................................28 Conceptual understanding of mathematics...................................................29 Procedural knowledge of mathematics.........................................................34 Procedural knowledge and doing mathematics............................................36 A constructivist approach to teaching the four operations..............................................38 Classroom exercises on the basic operations................................................38 The role of models in developing understanding............................................................43 Models for mathematical concepts...............................................................44 Using models in the teaching of place value................................................47 Models and constructing mathematics..........................................................51 Explaining the idea of a model.....................................................................53 Using models in the classroom.....................................................................54 Models in your classroom.............................................................................55 Strategies for effective teaching......................................................................................56 Unit Summary..................................................................................................................57 Self assessment................................................................................................................59 References.......................................................................................................................60

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Unit Two: Developing Understanding in MathematicsHow the unit fits into the moduleOverview of content of module The module Teaching and Learning Mathematics in Diverse Classrooms is intended as a guide to teaching mathematics for in-service teachers in primary schools. It is informed by the inclusive education policy (Education White Paper 6 Special Needs Education, 2001) and supports teachers in dealing with the diversity of learners in South African classrooms. In order to teach mathematics in South Africa today, teachers need an awareness of where we (the teachers and the learners) have come from as well as where we are going. Key questions are: Where will the journey of mathematics education take our learners? How can we help them? To help learners, we need to be able to answer a few key questions:  What is mathematics? What is mathematics learning and teaching in South Africa about today?  How does mathematical learning take place?  How can we teach mathematics effectively, particularly in diverse classrooms?  What is ‘basic’ in mathematics? What is the fundamental mathematical knowledge that all learners need, irrespective of the level of mathematics learning they will ultimately achieve?  How do we assess mathematics learning most effectively? These questions are important for all learning and teaching, but particularly for learning and teaching mathematics in diverse classrooms. In terms of the policy on inclusive education, all learners – whatever their barriers to learning or their particular circumstances in life – must learn mathematics. The units in this module were adapted from a module entitled Learning and Teaching of Intermediate and Senior Mathematics, produced in 2006 as one of the study guide for UNISA’s Advanced Certificate in Education programme. The module is divided into six units, each of which addresses the above questions, from a different perspective. Although the units can be studied separately, they should be read together to provide comprehensive guidance in answering the above questions. 1

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document. Unit 1: Exploring what it means to ‘do’ mathematics This unit gives a historical background to mathematics education in South Africa, to outcomes-based education and to the national curriculum statement for mathematics. The traditional approach to teaching mathematics is then contrasted with an approach to teaching mathematics that focuses on ‘doing’ mathematics, and mathematics as a science of pattern and order, in which learners actively explore mathematical ideas in a conducive classroom environment. Unit 2: Developing understanding in mathematics In this unit, the theoretical basis for teaching mathematics – constructivism – is explored. Varieties of teaching strategies based on constructivist understandings of how learning best takes place are described. Unit 3: Teaching through problem solving In this unit, the shift from the rule-based, teaching-by-telling approach to a problem-solving approach to mathematics teaching is explained and illustrated with numerous mathematics examples. Unit 4: Planning in the problem-based classroom In addition to outlining a step-by-step approach for a problem-based lesson, this unit looks at the role of group work and co-operative learning in the mathematics class, as well as the role of practice in problem-based mathematics classes. Unit 5: Building assessment into teaching and learning This unit explores outcomes-based assessment of mathematics in terms of five main questions – Why assess? (the purposes of assessment); What to assess? (achievement of outcomes, but also understanding, reasoning and problem-solving ability); How to assess? (methods, tools and techniques); How to interpret the results of assessment? (the importance of criteria and rubrics for outcomes-based assessment) ; and How to report on assessment? (developing meaningful report cards). Unit 6: Teaching all children mathematics This unit explores the implications of the fundamental assumption in this module – that ALL children can learn mathematics, whatever their background or language or sex, and regardless of learning disabilities they may have. It gives practical guidance on how teachers can adapt their lessons according to the specific needs of their learners. During the course of this module we engage with the ideas of three teachers - Bobo Diphoko, Jackson Segoe and Millicent Sekesi. Bobo, Jackson and Millicent are all teachers and close neighbours. Bobo teaches Senior Phase and Grade 10-12 Mathematics in the former Model C High School in town;2

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Unit Two: Developing Understanding in Mathematics Jackson is actually an Economics teacher but has been co-opted to teach Intermediate Phase Mathematics and Grade 10-12 Mathematical Literacy at the public Combined High School in the township; Millicent is the principal of a small farm-based primary school just outside town. Together with two other teachers, she provides Foundation Phase learning to an average 200 learners a year. Each unit in the module begins with a conversation between these three teachers that will help you to begin to reflect upon the issues that will be explored further in that unit. This should help you to build the framework on which to peg your new understandings about teaching and learning Mathematics in diverse classrooms. .How this unit is structured The unit consists of the following:  Welcome to the unit – from the three teachers who discuss their challenges and discoveries about mathematics teaching.  Unit outcomes.  Content of the unit, divided into sections.  A unit summary.  Self assessment.  References (sources used in the unit). 3

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document.Unit Two: Developing Understanding inMathematicsWelcome “I was thinking about our conversation last week,” said Millicent. “I remembered something I read a long time ago. The writers said that teaching and learning was a bit like building a bridge; we can provide the means and the support but the learners have to physically cross the bridge themselves – some will walk, some will run and some will need a lot of prompting to get to the other side.” “That sounds a bit philosophical to me,” remarked Bobo, “how does that help in practice?” “Well,” Millicent replied, “it helped me to understand that my learners learn in different ways; if I could understand how they thought about things I could probably help them better. Let me give you an example. I gave some of my learners the following problem: 26 – 18. This is how Thabo and Mpho responded: Thabo wrote T U4

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Unit Two: Developing Understanding in Mathematics 2 6 – 1 8 1 2Mpho wrote T U 1 2 1 6 – 1 8 2 8I then tried to work out what thinking process Thabo and Mpho had gonethrough to get to their answers and that helped me to work out how Icould help them.”“But that must take hours for the big classes we have,” responded Bobo.“Well, yes it can,” said Millicent, “but not everybody has problems allthe time and often I noticed that several learners had the same kind ofproblems so I could work with them separately while the rest were busywith something else. Then I started getting them to explain to each otherhow they had arrived at solutions to the problems I set them. I found thatoften as they explained their thinking process to somebody else, theyspotted errors themselves or discovered more efficient ways of doingthings without needing me to help.”Think about the following:1 Consider Thabo’s and Mpho’s responses to Millicent’s task. What seems to be the reasoning used by these two learners and how could you use this understanding to support them?2 Have you ever tried to get learners to explain to one another how they arrived at a particular solution to a problem? Can you suggest some potential advantages, disadvantages and alternatives to this approach?3 From her practice, what seems to be Millicent’s view of teacher and learner roles in developing understanding? Are you comfortable with this view? Why/why not?Comments:1 Thabo seems to have learned that you always take the smaller number away from the bigger number. Mpho seems to know the rule to ‘borrow’ from the tens and add to the units. Once that has been done, Mpho thinks she has completed the calculation. Now she just needs to complete the sum and since addition seems most natural she adds the 1 and 1 in the tens column to get 2. In both cases the learners are working through what they think is a correct formal process without regard to what the sums really mean. It might help to first get them to estimate the answers. They would also probably benefit from talking more about the processes they use in solving real-life problems and how these thinking processes can be captured in writing.2 Van Heerden (2003:30-31) points to the work of Resnick and Ford (1984) who remind us that “one of the fundamental assumptions of cognitive psychology is that the new knowledge is in large part constructed by the learner.” Getting children to talk through their reasoning with others helps them, their peers and you as the teacher to 5

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document. understand the assumptions and leaps of logic that learners make when in the process of constructing their own understanding. This unit will explore this process and how you can support it in more detail. 3 For Millicent, it would seem that the learners must be active participants in the meaning-making process. Using her wider experience she can be both guide and facilitator but she cannot simply transfer her own reasoning into the heads of her learners. This is in line with the major shift away from content-driven towards the more learning-driven approaches of OBE.Unit outcomes Upon completion of Unit Two you will be able to:  Critically reflect on the constructivist approach as an approach to learning mathematics.  Cite with understanding some examples of constructed learning as opposed to rote learning. Outcomes  Explain with insight the term understanding in terms of the measure of quality and quantity of connections.  Motivate with insight the benefits of relational understanding.  Distinguish and explain the difference between the two types of knowledge in mathematics: conceptual knowledge and procedural knowledge.  Critically discuss the role of models in developing understanding in mathematics (using a few examples).  Motivate for the three related uses of models in a developmental approach to teaching.  Describe the foundations of a developmental approach based on a constructivist view of learning.  Evaluate the seven strategies for effective teaching based on the perspectives of this chapter.6

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Unit Two: Developing Understanding in MathematicsIntroduction In recent years there has been an interesting move away from the idea that teachers can best help their learners to learn mathematics by deciding in what order and through what steps new material should be presented to learners. It has become a commonly accepted goal among mathematics educators that learners should be enabled to understand mathematics.  A widely accepted theory, known as constructivism, suggests that learners must be active participants in the development of their own understanding.  Each learner, it is now believed, constructs his/her own meaning in his/her own special way.  This happens as learners interact with their environment, as they process different experiences and as they build on the knowledge (or schema) which they already have. Njisane (1992) in Mathematics Education explains that learners never mirror or reflect what they are told or what they read: It is in the nature of the human mind to look for meaning , to find regularity in events in the environment whether or not there is suitable information available. The verb ‘to construct’ implies that the mental structures (schemas) the child ultimately possesses are built up gradually from separate components in a manner initially different from that of an adult. Constructivism derives from the cognitive school of psychology and the theories of Piaget and Vygotsky. It first began to influence the educational world in the 1960s. More recently, the ideas of constructivism have spread and gained strong support throughout the world, in countries like Britain, Europe, Australia and many others. Here in South Africa, the constructivist theory of mathematics learning has been strongly supported by researchers, by teachers and by the education department. The so-called Problem-centred Approach of Curriculum 2005 (which preceded the NCS) was implemented in the Foundation, Intermediate and Senior Primary phases in many South African schools. This was based on constructivist principles. The NCS also promotes a constructivist approach to teaching and learning mathematics, as you will have seen in unit one, particularly in the section where the action words of doing maths were discussed. Constructivism provides the teachers with insights concerning how children learn mathematics and guides us to use instructional strategies developmentally, that begin with the children and not ourselves. This chapter focuses on understanding mathematics from a constructivist perspective and reaping the benefits of relational understanding of mathematics, that is, linking procedural and conceptual knowledge to set the foundations of a developmental approach. 7

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document.A constructivist view of learning The constructivist view requires a shift from the traditional approach of direct teaching to facilitation of learning by the teacher. Teaching by negotiation has to replace teaching by imposition; learners have to be actively involved in ‘doing’ mathematics. This doing need not always be active and involve peer discussion, though it often does. Learners will also engage in constructive learning on their own, working quietly through set tasks, allowing their minds to sift through the materials they are working with, and consolidate new ideas together with existing ideas. Constructivism rejects the notion that children are blank slates with no ideas, concepts and mental structures. They do not absorb ideas as teachers present them, but rather, children are creators of their own knowledge. The question you should be asking now is: How are ideas constructed by the learners? The activity below will get you thinking about different ways in which teachers try to help learners construct their own understanding of key concepts. You should complete the activity according to your own experience as a mathematics learner and teacher.8

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Unit Two: Developing Understanding in Mathematics Constructing ideas Read through the following approaches that a teacher may employ to help learners to construct concepts, rules or principles.Activity 1 Instructing learners to memorize rules. Think about it for a while and then rate Explaining the rules /concepts to the each approach from 1 learners to 4 to indicate its effectiveness in Repetitive drilling of facts constructing /rules/principles meaningful ideas for the learner. Providing opportunities to learners to give expressions to their personal In the box next to the constructions. stated approach, write 1, 2, 3 or 4: Providing a supportive environment where learners feel free to share their 1 means that the initial conclusions and constructions. approach is not effective. Providing problem-solving approaches to enhance the learner construction of 2 means that the knowledge. approach is partially effective. Providing for discovery learning which results from the learner manipulating 3 means that the and structuring so that he or she finds approach is effective. new information. 4 means that the approach is very Using games to learn mathematics. effective. Reflection:  What criteria did you use in rating the above approaches?  In each of the above approaches, consider the extent to which all learners are involved in doing mathematics.  Is it possible that the different approaches could be weighted differently depending on the particular learning outcomes and context being explored? If so, can you give an example? 9

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document. The construction of ideas A key idea of constructivism is simply this: children construct their own knowledge. It is not just children who do this: everyone is involved all the time in making meaning and constructing their own understanding of the world. The constructivist approach views the learner as someone with a certain amount of knowledge already inside his or her head , not as an empty vessel which must be filled. The learner adds new knowledge to the existing knowledge by making sense of what is already inside his or her head . We, therefore, infer that the constructive process is one in which an individual tries to organize, structure and restructure his / her experiences in the light of available schemes of thought. In the process these schemes are modified or changed . Njisane (1992) explains that concepts, ideas, theories and models as individual constructs in the mind are constantly being tested by individual experiences and they last as long as they are interpreted by the individual. No lasting learning takes place if the learner is not actively involved in constructing his or her knowledge. Piaget (Farrell: 1980) insists that knowledge is active, that is, to know an idea or an object requires that the learner manipulates it physically or mentally and thereby transforms (or modifies) it. According to this concept, when you want to solve a problem relating to finance, in the home or at the garage or at the church, you will spontaneously and actively interact with the characteristics of the real situation that you see as relevant to your problem. A banker, faced with a business problem, may turn it over in his mind, he may prepare charts or look over relevant data, and may confer with colleagues − in so doing, he transforms the set of ideas in a combination of symbolic and concrete ways and so understands or knows the problem. The tools we use to build understanding are our existing ideas, the knowledge that we already possess. The materials we act on to build understanding may be things we see, hear or touch − elements of our physical world. Sometimes the materials are our own thoughts and ideas − to build our mental constructs upon. The effort that must be supplied by the learner is active and reflective thought. If the learners mind is not actively thinking, nothing happens. In order to construct and understand a new idea, you have to think actively about it. Mathematical ideas cannot be poured into a passive learner with an inactive mind. Learners must be encouraged to wrestle with new ideas, to work at fitting them into existing networks of ideas, and to challenge their own ideas and those of others. Van de Walle (2004) aptly uses the term reflective thought to explain how learners actively think about or mentally work on an idea. He says:10

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Unit Two: Developing Understanding in Mathematics Reflective thought means sifting through existing ideas to find those that seem to be the most useful in giving meaning to the new idea. Through reflective thought, we create an integrated network of connections between ideas (also referred to as cognitive schemas ). As we are exposed to more information or experience, the networks are added to or changed – so our cognitive or mental schemas are always being modified to include new ideas. Below is an example of the web of association that could contribute to the understanding of the concept ‘ratio’. DIVISION TRIGONOMETRY The ratio of 3 to 4 can be All trig functions are written as the division ratios relationship ¾. SCALE COMPARISON S The scale of a map is 1cm per 50km. We write The ratio of rainy this as the ratio 1:50 000 days to sunny days is greater in Cape RATIO Town than in the Great Karoo. SLOPE The ratio of the rise to the run is 1/8. UNIT PRICES 125g/R19,95. That’s R39,90 for 250g orR159,60 per kilogram. BUSINESS GEOMETRY Profit and loss are The ratio of the circumference of a figured as ratios of circle to its diameter is always . income to total cost. is approximately Any two similar figures have corresponding measurements that are proportional (in the same ratio). 11

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document. Cognitive schema: a network of connections between ideas. Select a particular skill (with operations, addition of fractions for example) that you would want your learners to acquire with understanding. Develop a cognitive schema (mental picture) for the newly Activity 2 emerging concept (or rule). You should consider the following:  Develop a network of connections between existing ideas (e.g. whole numbers, concept of a fraction, operations etc).  Add the new idea (addition of fractions for example). Draw in the connecting lines between the existing ideas and the new ideas used and formed during the acquisition of the skill. The general principles of constructivism are based largely on the work of Piaget. He says that when a person interacts with an experience/situation/idea, one of two things happens. Either the new experience is integrated into his existing schema (a process called assimilation ) or the existing schema has to be adapted to accommodate the new idea/experience (a process called adaptation ).  Assimilation refers to the use of an existing schema to give meaning to new experiences. Assimilation is based on the learner’s ability to notice similarities among objects and match the new ideas to those he/she already possesses.  Accommodation is the process of altering existing ways of seeing things or ideas that do not fit into existing schemata. Accommodation is facilitated by reflective thought and results in the changing or modification of existing schemata. The following activity will help you work through these ideas with a mathematical example.12

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Unit Two: Developing Understanding in Mathematics Daniel, a learner in grade 4, gives the following incorrect response 1 1 2 + = 2 2 4Activity 3 1 Explain the conceptual error made by the learner 2 What mental construct (or idea) needs to be modified by the learner to overcome this misconception? (Think of the addition of whole numbers and so on.) 3 Describe a useful constructive activity that Daniel could engage in to remedy the misconception. (He could use drawings, counters etc.) What kind of process takes place as a result of the modification of Daniels mental construct: accommodation or assimilation? Explain your answer. Implications for teaching Mathematics learning is likely to happen when we:  Use activities which build upon learners’ experiences  Use activities which the learners regard as powerful and interesting  Provide feedback to the learners  Use and develop correct mathematical language  Challenge learners within a supportive framework  Encourage learner collaboration, consensus and decision-making. Examples of constructed learning When learners construct their own conceptual understanding of what they are being taught, they will not always produce solutions that look the same. The teacher needs to be open to evaluating the solution of the learner as it has been presented. Computational proficiency and speed are not always the goal. Rather, confidence, understanding and a belief in their ability to solve a problem should also be valued. 13

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document. Consider the following two solutions to a problem . Case Study A Both solutions are correct and demonstrate conceptual understanding on behalf of the learners. An algorithm is a procedural method for doing a computation. Neither Michael nor Romy (above) has used formal division algorithms (such as long or short division). Subtraction using the vertical algorithm Activity 4 1 What calculation error did the learner make in subtraction? 2 What conceptual error did the learner make? (Think of place-value concepts). 3 Was the rule borrow from the next column clearly understood by the learner? Explain your answer. 4 In many instances, the learners existing knowledge is incomplete or inaccurate − so he/she invents an incorrect meaning. Explain the subtraction error in the light of the above statement.14

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Unit Two: Developing Understanding in MathematicsConstruction in rote learningAll that you have read so far shows that learning and thinking cannot beseparated from each other (especially in mathematics). In manyclassrooms, reflective thought (or active thinking) is still often replacedby rote learning with the focus on the acquisition of specific skills, factsand the memorizing of information, rules and procedures, most of whichis very soon forgotten once the immediate need for its retention is passed.A learner needs information, concepts, ideas, or a network of connectedideas in order to think and he will think according to the knowledge healready has at his disposal (in his cognitive schemata). The dead weightof facts learnt off by heart, by memory without thought to meaning (thatis rote learning), robs the learner of the potential excitement of relatingideas or concepts to one another and the possibility of divergent andcreative thinking (Grossmann: 1986).Constructivism is a theory about how we learn. So, even rote learning is aconstruction. However, the tools or ideas used for this construction inrote learning are minimal. You may well ask: To what is knowledgelearned by rote, connected?What is inflicted on children as a result of rote-memorized rules, in manycases, is the manipulation of symbols, that have little or no attachedmeaning.This makes learning much more difficult because rules are much harderto remember than integrated conceptual structures which are made up ofa network of connected ideas. In addition, careless errors are not pickedup because the task has no meaning for the learner and so he/she has notanticipated the kind of result that might emerge.According to the stereotypical traditional view, mathematics is regardedas a “tool subject” consisting of a series of computational skills: the rotelearning of skills is all-important with rate and accuracy the criteria formeasuring learning. This approach, labelled as the drill theory, wasdescribed by William Brawnell (Paul Trapton: 1986) as follows: Arithmetic consists of a vast host of unrelated facts and relatively independent skills. The pupil acquires the facts by repeating them over and over again until he is able to recall them immediately and correctly. He develops the skills by going through the processes in question until he can perform the required operations automatically and accurately. The teacher need give little time to instructing the pupil in the meaning of what he is learning.There are numerous weaknesses with this approach: Learners perform poorly, neither understanding nor enjoying the subject; They are unable to apply what they have learned to new situations; they soon forget what they have learned; 15

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document.  Learning occurs in a vacuum; the link to the real world is rarely made;  Little attention is paid to the needs, interest and development of the learner;  Knowledge learned by rote is hardly connected to the child’s existing ideas (that is, the childs cognitive schemata) so that useful cognitive networks are not formed - each newly-formed idea is isolated;  Rote learning will almost never contribute to a useful network of ideas.  Rote learning can be thought of as a weak construction. Rote learning An enthusiastic class in the Senior Phase is put through a rigorous process of rote learning in mathematics. Activity 5 1 From an OBE perspective, would you approve of this approach? Explain your response. List your major concerns with regard to effective and meaningful learning taking place in this class. 2 Set a task for your learners to think of creative ways to remember that 7 × 8 = 56. They should create their own useful mathematical networks.  (You could engage your learners in the Intermediate Phase with this activity). Consider the clever ways the class figured out the product.  Now explain some clever ways the class could use to remember that 16 × 25 = 400.  Compare the memorization of these facts (7 × 8 = 56 or 6 × 9 = 54) by rote to the network of profitable mental constructions (leading to these products). Which approach would you prefer? Explain your response briefly. Do you think that ultimately all senior phase learners should have the multiplication tables at their fingertips? Give reasons for your answers. Understanding We are now in a position to say what we mean by understanding . Grossman (1986) explains that to understand something means to assimilate it into an appropriate schema (cognitive structure). Recall that assimilation refers to the use of an existing schema (or a network of connected ideas) to give meaning to new experiences and new ideas. It is important to note that the assimilation of information or ideas to an16

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Unit Two: Developing Understanding in Mathematicsinappropriate (faulty, confusing, or incorrect) schema will make theassimilation to later ideas more difficult and in some cases perhapsimpossible (depending on how inappropriate the schema is).Grossmann (1986) cites another obstacle to understanding: the belief thatone already understands fully - learners are very often unaware that theyhave not understood a concept until they put it into practice. How oftenhas a teacher given a class a number of similar problems to do (afterdemonstrating a particular number process on the board) only to find anumber of children who cannot solve the problems? Those childrenthought that they understood, but they did not. The situation becomes justas problematic when there is an absence of a schema: that is, no schemato assimilate to, just a collection of memorised rules and facts. Forteachers in the intermediate phase the danger lies in the fact thatmechanical computation can obscure the fact that schemata are not beingconstructed or built up, especially in the first few years – this is to thedetriment of the learners’ understanding in later years.Understanding can be thought of as the measure of the quality andquantity of connections that an idea has with existing ideas.Understanding depends on the existence of appropriate ideas and thecreation of new connections. The greater the number of appropriateconnections to a network of ideas, the better the understanding will be. Aperson’s understanding exists along a continuum . At one pole, an idea isassociated with many others in a rich network of related ideas. This is thepole of so-called ‘relational understanding’. At the other, the ideas areloosely connected, or isolated from each other. This is the pole of so-called ‘instrumental understanding’.Knowledge learned by rote is almost always at the pole of instrumentalunderstanding - where ideas are nearly always isolated and disconnected.Grossman (1986) draws attention to one of Piaget’s teaching and learningprinciples: the importance of the child learning by his or her owndiscovery. When learners come to knowledge through self discovery , theknowledge has more meaning because discovery facilitates the process ofbuilding cognitive structures (constructing a network of connected ideas).Recall of information (concepts, procedures) is easier than recall ofunrelated knowledge transmitted to the learner.Through the process of discovery (or investigation), a learner passesthrough a process of grasping the basic relations (or connections) of anevent while discarding irrelevant relations and so he or she arrives at aconcept (idea) together with an understanding of the relations that givethe concept meaning: the learner can, therefore, go on to handle and copewith a good deal of meaningful new, but in fact highly relatedinformation.We infer from the above that the learner arrives at a concept that isderived from a schema (a network of connected ideas) rather than fromdirect instruction from the teacher. This produces the kind of learner whois independent, able to think, able to express ideas, and solve problems.This represents a shift to learner centeredness − where learners areknowledge developers and users rather than storage systems andperformers (Grossman: 1986). 17

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document. Relational and instrumental understanding 1 Explain the difference between relational understanding and instrumental understanding. Activity 6 2 Explain why relational understanding has a far greater potential for promoting reflective thinking than instrumental understanding . 3 Explain what it means to say that understanding exists on a continuum from relational to instrumental. Give an example of a mathematical concept and explain how it might be understood at different places along this continuum. Explain the difference between the use of a rule, an algorithm or an idea, and the thinking strategies, processes and concepts used to construct a solution (or a new idea). Examples of understanding Understanding is about being able to connect ideas together, rather than simply knowing isolated facts. The question Does the learner know it? must be replaced with How well does the learner understand it? The first question refers to instrumental understanding and the second leads to relational understanding. Memorising rules and using recipe methods diligently in computations is knowing the idea. Where the learner connects a network of ideas to form a new idea and arrive at solutions, this is understanding the idea and contributes to how a learner understands. Let’s illustrate this with an example. Look at the subtraction skill involved in the following: 15 −6 . Reflect on the thought processes at different places along the understanding continuum (that is, the continuous closing of gaps for the understanding of the idea at hand).18

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Unit Two: Developing Understanding in MathematicsThis kind of analysis of steps that learners could follow when answeringa question can help you, the teacher, to help learners to overcome theirdifficulties and misunderstandings. There is not a ‘magic wand’ that onecan wave to make problems go away. Each individual learner will needattention and help at the point at which he/she is experiencing difficulty,and you need to be able to find the step at which he/she needs help, andtake it from there.Benefits of relational understandingReflect again on the involvement of the learner in the science of patternand order when ‘doing’ mathematics. Perhaps he or she had to shareideas with others, whether right or wrong, and try to defend them. Thelearner had to listen to his/her peers and try to make sense of their ideas.Together they tried to come up with a solution and had to decide if theanswer was correct without looking in an answer book or even asking theteacher. This process takes time and effort.When learners do mathematics like this on a daily basis in anenvironment that encourages risk and participation, formulating anetwork of connected ideas (through reflecting, investigating and problemsolving ), it becomes an exciting endeavour, a meaningful andconstructive experience.In order to maximise relational understanding, it is important for theteacher to select effective tasks and mathematics activities that lend themselves to exploration, investigation (of number patterns for example) or self- discovery; make instrumental material available (in the form of manipulatives, worksheets, mathematical games and puzzles, diagrams and drawings, paper-folding, cutting and pasting, and so on) so that the learners can engage with the tasks; organise the classroom for constructive group work and maximum interaction with and among the learners.The important benefits derived from relational understanding (that is, thismethod of constructing knowledge through the process of doing mathematics in problem-solving and thus connecting a network of ideasto give meaning to a new idea) make the whole effort not onlyworthwhile but also essential.In his book Elementary and Middle School Mathematics, Van de Walle(2004) gives a very clear account of seven benefits of relationalunderstanding (see Chapter Three). What following is a slightly adaptedversion of his account. 21

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document. Benefit 1: It is intrinsically rewarding Nearly all people, and certainly children, enjoy learning (what type of learning?, you may ask). This is especially true when new information, new concepts and principles connect with ideas already at the learner’s disposal. The new knowledge now makes sense, it fits (into the learner’s schema) and it feels good. The learner experiences an inward satisfaction and derives an inward motivation to continue, to search and explore further - he or she finds it intrinsically rewarding. Children who learn by rote (memorisation of facts and rules without understanding) must be motivated by external means: for the sake of a test, to please a parent, from fear of failure, or to receive some reward. Such learning may not result in sincere inward motivation and stimulation. It will neither encourage the learner nor create a love for the subject when the rewards are removed. Benefit 2: It enhances memory Memory is a process of recalling or remembering or retrieving of information. When mathematics is learned relationally (with understanding) the connected information , or the network of connected ideas, is simply more likely to be retained over time than disconnected information. Retrieval of information is also much easier. Connected information provides an entire web of ideas (or network of ideas). If what you need to recall seems distant, reflecting on ideas that are related can usually lead you to the desired idea eventually. Retrieving disconnected information or disorganised information is more like finding a needle in haystack. Look at the example given below. Would it be easier to recall the set of disconnected numbers indicated in column A, or the more organized list of numbers in column B? Does the identification of the number pattern in column B (that is, finding a rule that connects the numbers) make it easier to retrieve this list of numbers? A B 9 5 15 1 3 5 7 9 11 13 15 17 19 7 19 17 3 11 1 13 Organised and connected list of numbers Disconnected list of numbers22

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Unit Two: Developing Understanding in Mathematics Benefit 3: There is less to remember Traditional approaches have tended to fragment mathematics into seemingly endless lists of isolated skills, concepts, rules and symbols. The lists are so lengthy that teachers and learners become overwhelmed from remembering or retrieving hosts of isolated and disconnected information . Constructivists, for their part, talk about how ‘big ideas’ are developed from constructing large networks of interrelated concepts. Ideas are learned relationally when they are integrated into a web of information, a ‘big idea’. For a network of ideas that is well constructed, whole chunks of information are stored and retrieved as a single entity or as a single ground of related concepts rather than isolated bits. Think of the big idea ‘ratio and proportion’ and how it connects and integrates various aspects of the mathematics curriculum: the length of an object and its shadow, scale drawings, trigonometric ratios, similar triangles with proportional sides, the ratio between the area of a circle and its radius and so on. Another example - knowledge of place value - underlies the rules involving decimal numbers: EXAMPLE Lining up decimal numbers 53,25 0,37Case Study B +8,01 …….. Ordering decimal numbers 8,45 ; 8,04 ; 8,006 (in descending order) Decimal-percent conversions 85 0,85 = = 85% 100 Rounding and estimating Round off 84,425 to two decimal places Answer: 84,43 Converting to decimal 1 • = 0.3333... or 0.3 3 Converting to fractions 75 3 0,75 kg = kg = kg 100 4 and so on. 23

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Unit Two: Developing Understanding in MathematicsLeads to:3x + 4x = 7xTake careful note of how connections are made and new constructs orideas are generated. Without these connections, learners will need tolearn each new piece of information they encounter as a separateunrelated idea.Benefit 5: It improves problem-solving abilitiesThe solution of novel problems (or the solution of problems that are notthe familiar routine type) requires transferring ideas learned in onecontext to new situations. When concepts, skills or principles areconstructed in a rich and organised network (of ideas), transferability to anew situation is greatly enhanced and, thus, so is problem solving.Consider the following example:Learners in the Intermediate phase are asked to work out the followingsum in different ways:14 + 14 + 14 + 14 + 14 + 14 + 14 + 6 + 6 + 6 + 6 + 6 + 6 + 6Learners with a rich network of connected ideas with regard to theaddition of whole numbers, multiplication as repeated addition and theidentification of number patterns might well construct the followingsolutions to this problem:7 × (14 + 6) = 7 × 20 (since there are seven pairs of the sum 14 + 6) = 140or7 × 14 + 7 × 6 (seven groups of 14 and seven groups of 6)= 98 + 42= 140Adding the numbers from left to right would be, you must agree, atedious exercise.Benefit 6: It is self-generativeA learner who has constructed a network of related or connected ideaswill be able to move much more easily from this initial mental state to anew idea, a new construct or a new invention. This learner will be able tocreate a series of mental pathways, based on the cognitive map ofunderstanding (a rich web of connected ideas) at his or her disposal, to anew idea or solution. 25

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document. That is, the learner finds a path to a new goal state. Van de Walle agrees with Hiebert and Carpenter that a rich base of understanding can generate new understandings: Inventions that operate on understanding can generate new understanding, suggesting a kind of snowball effect. As networks grow and become more structured, they increase the potential for invention. Consider as an example, the sum of the first ten natural numbers: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 A learner with insight and understanding of numbers may well realize that each consecutive number from left to right increases by one and each consecutive number from right to left decreases by one. Hence the following five groups of eleven are formed: 1 + 10 = 11; 2 + 9 =11; 3 + 8 =11; 4 + 7 =11 and 5 + 6 =11 The sum of the first ten natural numbers is simply: (10 + 1) × 10 = 11 × 5 = 55 2 The connection of ideas in the above construct may well generate an understanding of the following new rule: n 1 + 2+ 3 + 4+...........+ n = ( n + 1) × 2 Use this rule to add up the first 100 natural numbers. Relational understanding therefore has the welcome potential to motivate the learner to new insights and ideas, and the creation of new inventions and discoveries in mathematics. When gaining knowledge is found to be pleasurable, people who have had that experience of pleasure are more likely to seek or invent new ideas on their own, especially when confronting problem-based situations. Benefit 7: It improves attitudes and beliefs Relational understanding has the potential to inspire a positive feeling, emotion or desire (affective effect) in the learner of mathematics, as well as promoting his or her faculty of knowing, reasoning and perceiving (cognitive effect). When learning relationally, the learner tends to develop a positive self-concept, self-worth and confidence with regard to his or her ability to learn and understand mathematics.26

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Unit Two: Developing Understanding in MathematicsRelational understanding is the product of a learning process wherelearners are engaged in a series of carefully designed tasks which aresolved in a social environment .Learners make discoveries for themselves, share experiences with others,engage in helpful debates about methods and solutions, invent newmethods, articulate their thoughts, borrow ideas from their peers andsolve problems − in so doing, conceptual knowledge is constructed andinternalised by the learner improving the quality and quantity of thenetwork of connected and related ideas.The effects may be summarized as follows: promotes self-reliance and self-esteem promotes confidence to tackle new problems reduces anxiety and pressure develops an honest understanding of concepts learners do not rely on interpretive learning but on the construction of knowledge learners develop investigative and problem-solving strategies learners do not forget knowledge they have constructed learners enjoy mathematics.There is no reason to fear or to be in awe of knowledge learnedrelationally. Mathematics now makes sense - it is not some mysteriousworld that only ‘smart people’ dare enter. At the other end of thecontinuum, instrumental understanding has the potential of producingmathematics anxiety, or fear and avoidance behaviour towardsmathematics.Relational understanding also promotes a positive view aboutmathematics itself. Sensing the connectedness and logic of mathematics,learners are more likely to be attracted to it or to describe it in positiveterms.In concluding this section on relational understanding, let us remindourselves that the principles of OBE make it clear that learning withunderstanding is both essential and possible. That is, all learners can andmust learn mathematics with understanding. Learning with understandingis the only way to ensure that learners will be able to cope with the manyunknown problems that will confront them in the future. 27

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document. Benefits of relational understanding Examine the seven benefits of relational understanding given above. Select the benefits that you think are most important for the learning (with Activity 7 understanding) of mathematics. Discuss this with fellow mathematics teachers. Describe each of the benefits chosen above and then explain why you personally believe each one is significant. Try to illustrate your thinking with a practical example from your own classroom.Types of mathematical knowledge All knowledge, whether mathematical or other knowledge, consists of internal or mental representations of ideas that the mind has constructed. The concept itself, then, exists in the mind as an abstraction . To illustrate this point Farrell and Farmer (1980) refer to the formation of the concept of a triangle as follows: When you learned the concept of triangle you may have been shown all kinds of triangular shapes like cardboard cut-outs, three pipe cleaners tied together, or pictures of triangular structures on bridges or pictures of triangle in general. Eventually, you learned that these objects and drawings were representations or physical models of a triangle, not the triangle itself. In fact, you probably learned the concept of triangle before you were taught to give a definition and you may have even learned quite a bit about the concept before anyone told you its name. So a concept is not its label, nor is it any physical model or single example. The concept of the triangle, therefore, resides in the mental representation of the idea that the mind has constructed. You may also include terms such as integer, pi (π), locus, congruence, set addition, equality and inequality as some of the mental representations of ideas that the mind has constructed in mathematics. According to Njisane (1992) in Mathematics Education, Piaget distinguishes three types of knowledge, namely, social, physical and logico-mathematical knowledge:  Social knowledge is dependent on the particular culture. In one culture it is accepted to eat with ones fingers, in another it may be considered as bad manners. Social knowledge is acquired through interaction with28

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Unit Two: Developing Understanding in Mathematics other people. Presumably the best way to teach it in the classroom would be through telling.  Physical knowledge is gained when one abstracts information about the objects themselves. The colour of an object, its shape, what happens to it when it is knocked against a wall and so on are examples of physical knowledge.  Logico-mathematical knowledge is made up of relationships between objects, which are not inherent in the objects themselves but are introduced through mental activity. For example, to acquire a concept of the number 3, a learner needs to experience different situations where three objects or elements are encountered. Logico- mathematical knowledge is acquired through reflective abstraction , depending on the child’s mind and the way he or she organizes and interprets reality. It seems that each one of us arrives at our own logico-mathematical knowledge. It is important to note that the acquisition of logico-mathematical knowledge without using social and physical knowledge as a foundation is bound to be ineffective. Since relational understanding depends on the integration of ideas into abstract networks of ideas (or a network of interconnected ideas), teachers of mathematics may just view mathematics as something that exists out there, while forgetting the concrete roots of mathematical ideas. This could result in a serious mistake - teachers must take into account how these mental representations of the mind are constructed. That is, through effective interaction and doing mathematics. Logico-mathematical knowledge Read the text above and then answer the following question: Can logico-mathematical knowledge be transmitted from the teacher toActivity 8 the learner while the learner plays a passive role? Explain your response. In your response remember to consider that:  mental ideas or representations need to be constructed by the learner.  the learner requires interaction and mental activity to establish relationships between objects. Conceptual understanding of mathematics Conceptual knowledge of mathematics consists of logical relationships constructed internally and existing in the mind as a part of the greater network of ideas: 29

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document.  It is the type of knowledge Piaget referred to as logico-mathematical knowledge. That is, knowledge made up of relationships between objects, which are not inherent in the objects themselves, but are introduced through mental activity.  By its very nature, conceptual knowledge is knowledge that is understood. You have formed many mathematical concepts. Ideas such as seven, nine, rectangle, one/tens/hundreds (as in place value), sum, difference, quotient, product, equivalent, ratio, positive, negative are all examples of mathematical relationships or concepts. It would be appropriate at this stage to focus on the nature of mathematical concepts. Richard R Skemp (1964) emphasises the following (which should draw your attention): Mathematics is not a collection of facts, which can be demonstrated, seen or verified in the physical world (or external world), but a structure of closely related concepts, arrived at by a process of pure thought. Think about this: that the subject matter of mathematics (or the concepts and relationships) is not to be found in the external world (outside the mind), and is not accessible to our vision, hearing and other sense organs. These mathematical concepts have only mental existence - so in order to construct a mathematical concept or relationship, one has to turn it away from the physical world of sensory objects to an inner world of purely mental objects. This ability of the mind to turn inwards on itself, that is, to reflect , is something that most of us use so naturally that we may fail to realize what a remarkable ability it is. Do you not consider it odd that we can hear our own verbal thoughts and see our own mental images, although no one has revealed any internal sense organs which could explain these activities? Skemp (1964) refers to this ability of the mind as reflective intelligence. Are mathematical concepts different from scientific concepts? Farrell and Farmer (1980) explains that unlike other kinds of concepts such as cow, dog, glass, ant, water, flower, and the like, you cannot see or subject to the other senses examples of triangle, points, pi, congruence, ratio, negative numbers and so on. ‘But we write numbers, dont we?’ you may ask. No, we write symbols which some prefer to call numerals , the names for numbers. Now reflect on the following key difference between mathematics and science: Scientific concepts include all those examples which can be perceived by the senses, such as insect and flower, and those whose examples cannot be perceived by the senses, such as atom and gravity. These latter concepts are taught by using physical models or representations of the concepts (as in the case of mathematics). (Farrell & Farmer: 1980). Skemp (1964) urges us to see that the data of sensori-motor learning are sense data present in the external world - however, the data for reflective30

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Unit Two: Developing Understanding in Mathematicsintelligence are concepts, so these must have been formed in the learner’sown mind before he or she could reflect on them. A basic question thatyou may ask at this stage is: How are mathematical concepts formed?Skemp (1964) points out that to give someone a concept in a field ofexperiences which is quite new to him or her, we must do two things:Arrange for him or her a group of experiences which have the concept ascommon and if it is a secondary concept (that is a concept derived fromthe primary concepts), we also have to make sure that he or she has theother concepts from which it is derived (that is the prerequisite conceptsneed to be in place in the mental schema of the learner).Returning now to mathematics: ‘seven’ is a primary concept , representingthat which all collections of seven objects have in common. ‘Addition’ isanother concept, derived from all actions or processes which make twocollections into one. These concepts require for their learning a variety ofdirect sensory experiences (counters, manipulative and so on) from theexternal world to exemplify them.The weakness of our present teaching methods comes, according toSkemp (1964), during and after the transition from primary to secondaryconcepts, and other concepts in the hierarchy. For example, from workingthrough the properties of individual numbers to generalization aboutthese properties ; from statements like9 × 6 = 54 to those like 9(x + y) = 9x + 9y.Do you agree that many learners never do understand what thesealgebraic statements really mean, although they may, by rote-learning,acquire some skills in performing as required certain tricks with thesymbols? Understanding these statements requires the formation orconstruction of the appropriate mathematical concepts.Are there any limitations in the understanding of mathematical conceptslearnt through the use of physical objects and concrete manipulativesfrom the external world? Skemp (1964) agrees fully with Diennes that toenable a learner to form a new concept, we must give him (or her) anumber of different examples from which to form the concept in his orher own mind - for this purpose some clever and attractive concreteembodiments (or representations) of algebraic concepts in the form ofbalances, peg-boards, coloured shapes and frames and the like areavailable.However, these concrete embodiments fail to take into account theessential difference between primary and higher order concepts - that is,only primary concepts can be exemplified in physical or concrete objects,and higher order concepts can only be symbolised .To explain this, think of the concept 3 + 4 = 7, which can bedemonstrated physically with three blocks and four blocks or with beadsor coins, But3x + 4x = 7x 31

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document. is a statement that generalises what is common to all statements such as 3×5+4×5=7×5 3 × 8 + 4 × 8 = 7 × 8 etc. and which ignores particular results such as: 3 × 5 + 4 × 5 = 35. Do you agree, therefore, that understanding of the algebraic statement is derived from a discovery of what is common to all arithmetical statements of this kind, not of what is common to any act or actions with physical objects? As new concepts and relationships are being assimilated in the network of connected ideas, the direction of progress is never away from the primary concepts. This progress results in the dependence of secondary concepts upon primary concepts. Once concepts are sufficiently well formed and independent of their origins, they become the generators of the next higher set - and in so doing lead to the construction of a hierarchy of concepts. Van de Walle (2004: 26) cautions us that the use of physical (or concrete) objects in teaching may compromise meaningful understanding of concepts. This happens if there are insufficient opportunities for the learner to generalise the concept: Diennes’ blocks are commonly used to represent ones, tens and hundreds. Learners who have seen pictures of these or have used actual blocks may labour under the misconception that the rod is the ten piece and the large square block is the hundreds piece. Does this mean that they have constructed the concepts of ten and hundred? All that is known for sure is that they have learned the names for these objects, the conventional names of the blocks. The mathematical concept of ten is that a ten is the same as ten ones. Ten is not a rod. The concept is the relationship between the rod and the small cube - the concept is not the rod or a bundle of ten sticks or any other model of a ten. This relationship called ten must be created by learners in their own minds. Here is another interesting example that distinguishes the concept from the physical object. In this example the shapes are used to represent wholes and parts of wholes, in other words, this is an example dealing with the concept of a fraction. Reflect carefully on the three shapes (A, B and C) which can be used to represent different relationships.32

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Unit Two: Developing Understanding in Mathematics A B CIf we call shape B one or a whole, then we might refer to shape A asone-half . The idea of half is the relationship between shapes A and B, arelationship that must be constructed in our mind as it is not in therectangle.If we decide to call shape C the whole, shape A now becomes one-fourth. The physical model of the rectangle did not change in any way.You will agree that the concepts of half and fourth are not in rectangleA - we construct them in our mind . The rectangles help us to see therelationship, but what we see are rectangles, not concepts. Assigningdifferent rectangles the status of the ‘whole’ can lead to generalisation ofthe concept. 33

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Unit Two: Developing Understanding in Mathematics Error! No textof specified style in document. The formation of concepts For this activity you are required to reflect on conceptual knowledge in mathematics. Activity 9 1 Richard R Skemp states that mathematics is not a collection of facts which can be demonstrated and verified in the physical world, but a structure of closely related concepts, arrived at by a process of pure thought.  Discuss the above statement critically with fellow teachers of mathematics. Take into account how concepts and logical relationships are constructed internally and exist in the mind as part of a network of ideas.  In the light of the above statement explain what Skemp means when he refers to reflective intelligence (the ability of the mind to turn inwards on itself).  Why are scientific concepts different from mathematical concepts? Explain this difference clearly using appropriate examples. 2 Skemp points out that to help a learner construct a concept in a field of experience which is quite new to him or her, we must do two things. Mention the two activities that the teacher needs to follow to help the learner acquire primary concepts, secondary concepts, and other concepts in the hierarchy of concepts. 3 Skemp distinguishes between primary concepts and secondary concepts in the learning of mathematics. Reflect on the difference between concepts which are on different levels. Name some secondary concepts that learners in the Senior Phase may encounter. 4 Analyse the three shapes (A, B and C) shown in the text on the previous page. Explain why the concepts ‘half’ and ‘quarter’ are not physically in rectangle A - but in the mind of the learner. Explain the implications of this for teaching using manipulatives (concrete apparatus). Procedural knowledge of mathematics Procedural knowledge of mathematics is knowledge of the rules and procedures that one uses in carrying out routine mathematical tasks. It includes the symbolism that is used to represent mathematics. You could, therefore, infer that knowledge of mathematics consists of more than concepts. Step-by-step procedures exist for performing tasks such as: 56 × 74 (Multiplying two digit numbers) 1 932 ÷ 28 (Long division)34